Ebert's asymmetric three person three color Hat Game

TL;DR

This paper extends Ebert's three-person, three-color hat game to asymmetric probabilities, introducing the concept of adequate sets to optimize winning strategies efficiently.

Contribution

It introduces the use of adequate sets for analyzing asymmetric hat problems, reducing computational complexity and providing a systematic approach to strategy design.

Findings

Optimal strategies depend on asymmetric color probabilities.

Adequate sets simplify the analysis compared to standard methods.

The approach is probability-independent, aiding in diverse scenarios.

Abstract

We generalize Ebert's Hat Problem for three persons and three colors. All players guess simultaneously the color of their own hat observing only the hat colors of the other players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his or her hat color correct and none of the players has an incorrect guess. This paper studies Ebert's hat problem, where the probabilities of the colors may be different (asymmetric case). Our goal is to maximize the probability of winning the game and to describe winning strategies. In this paper we use the notion of an adequate set. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of the asymmetric case. Another point of interest is the fact that computational complexity using adequate sets is much less than using standard methods.